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Any data which are directly or indirectly referenced to a location on the surface of the earth (latitude & longitude values) are spatial data. Statistical data which deals with the spatial data is known as spatial statistics.

Spatial statistics & Classical statistics

Spatial Statistics- Mean centre

The mean is an important measure of central tendency for a set of data. If this concept of central tendency is extended to locational point data in two dimensions (X and Y coordinates), the average location, called the mean centre, can be determined.

Once the coordinate system has been established and the coordinates of each point determined the mean center can be calculated by separately averaging the X and Y coordinates, as follows:

Spatial measures of dispersion

Standard distance is the spatial equivalent of standard deviation. Standard distance measures the amount of absolute dispersion in a point pattern. After the location coordinates of the mean center have determined, the standard distance statistic incorporates the straight-line or Euclidean distance of each point from the center. Standard distance (SD)

Relative Distance

The coefficient of variation (standard deviation divided by the mean) is the classical measure of relative dispersion. A perfect spatial analogue to the coefficient of variation does not exist for measuring relative dispersion.

To derive a descriptive measure of relative spatial dispersion, the standard distance of a point pattern is divided by some measure of regional magnitude. One possible divisor is the radius (RA) of a circle with the same area as the region being analyzed. A useful measure of relative dispersion (RD) is given by

RD = SD/ RA

Spatial autocorrelation

Spatial autocorrelation statistics measure and analyze the degree of dependency among observations in a geographic space. Spatial autocorrelation may be classified as either positive or negative. Positive spatial autocorrelation has all similar values appearing together i.e. a map pattern where geographic features of similar value tend to cluster on a map. Negative spatial autocorrelation has dissimilar values appearing in close association i.e. a map pattern in which geographic units of similar values scatter throughout the map. When no statistically significant spatial autocorrelation exists, the pattern of spatial distribution is considered random.

One of the classic spatial autocorrelation statistics is Moran's . It is a weighted correlation co-efficient used to detect departures from spatial randomness. Departures from randomness indicate spatial patterns such as clusters.

The test statistic I, proposed by Moran is defined as

It is very similar to the product moment correlation co-efficient, except for the addition of the weight terms (wij). The weights reflect how connected the two areas are, and usually reflect the geographic proximity. Moran’s I is smaller than its expectation when the rates/values in connected areas are dissimilar. Weights quantify the hypothesis about how similar the rates in the different areas ought to be.

Under the null hypothesis of spatial random data, the mean and variance of I is

Where data with a random spatial distribution give an expected value of I close to zero, spatial aggregation or clustering leads to positive values, with an upper limit of one for extreme clustering